Question: $\dfrac{ 7d + 9e }{ 2 } = \dfrac{ -2d + 2f }{ -8 }$ Solve for $d$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 7d + 9e }{ {2} } = \dfrac{ -2d + 2f }{ -8 }$ ${2} \cdot \dfrac{ 7d + 9e }{ {2} } = {2} \cdot \dfrac{ -2d + 2f }{ -8 }$ $7d + 9e = {2} \cdot \dfrac { -2d + 2f }{ -8 }$ Multiply both sides by the right denominator. $7d + 9e = 2 \cdot \dfrac{ -2d + 2f }{ -{8} }$ $-{8} \cdot \left( 7d + 9e \right) = -{8} \cdot 2 \cdot \dfrac{ -2d + 2f }{ -{8} }$ $-{8} \cdot \left( 7d + 9e \right) = 2 \cdot \left( -2d + 2f \right)$ Distribute both sides $-{8} \cdot \left( 7d + 9e \right) = {2} \cdot \left( -2d + 2f \right)$ $-{56}d - {72}e = -{4}d + {4}f$ Combine $d$ terms on the left. $-{56d} - 72e = -{4d} + 4f$ $-{52d} - 72e = 4f$ Move the $e$ term to the right. $-52d - {72e} = 4f$ $-52d = 4f + {72e}$ Isolate $d$ by dividing both sides by its coefficient. $-{52}d = 4f + 72e$ $d = \dfrac{ 4f + 72e }{ -{52} }$ All of these terms are divisible by $4$ Divide by the common factor and swap signs so the denominator isn't negative. $d = \dfrac{ -{1}f - {18}e }{ {13} }$